Power integral bases in a family of sextic fields with quadratic subfields

TL;DR

This paper investigates power integral bases in a parametric family of sextic fields constructed from imaginary quadratic fields, providing explicit generators and classifications through relative Thue equations.

Contribution

It determines the generators of power integral bases in a broad family of sextic fields with quadratic subfields using relative Thue equations.

Findings

Explicit description of power integral bases in the family of sextic fields.

Identification of generators for absolute power integral bases.

Application of relative Thue equations to classify bases.

Abstract

Let $M=Q(i\sqrt{d})$ be any imaginary quadratic field with a positive square-free $d$. Consider the polynomial \[ f(x)=x^3-ax^2-(a+3)x-1, \] with a parameter $a\in Z$. Let $K=M(\alpha)$, where $\alpha$ is a root of $f$. This is an infinite parametric family of sextic fields depending on two parameters, $a$ and $d$. Applying relative Thue equations we determine the relative power integral bases of these sextic fields over their quadratic subfields. Using these results we also determine generators of (absolute) power integral bases of the sextic fields.